本文選題：實(shí)值離散Gabor變換　切入點(diǎn)：雙正交性　出處：《安徽大學(xué)》2016年博士論文

【摘要】：Gabor變換是重要的時頻分析方法之一,廣泛應(yīng)用于非平穩(wěn)信號的檢測、分析與處理。然而由于Heisenberg不確定原理的制約,傳統(tǒng)的單窗Gabor變換的時頻局域性(或者時頻分辨精度)受到很大限制。由于分析窗和綜合窗函數(shù)寬度是固定的,由單窗Gabor變換獲得的時頻譜時間分辨精度與頻率分辨精度也是固定的,并且不可能同時都好,二者是矛盾的。采用寬窗將產(chǎn)生高頻率分辨率但低時間分辨率的時頻譜；反之,使用窄窗將產(chǎn)生高時間分辨率但低頻率分辨率的時頻譜。于是學(xué)者們在框架理論基礎(chǔ)上提出了多窗復(fù)值離散Gabor變換(M-CDGT),通過組合多窗情況下獲得的時頻譜,可有效改善時頻表示精度。但M-CDGT的研究還不夠完善和深入,為了拓展多窗離散Gabor變換的研究,本文利用雙正交分析法研究了多窗實(shí)值離散Gabor變換(M-RDGT)。主要研究內(nèi)容和創(chuàng)新成果如下：提出了有限長序列M-RDGT及其快速算法。將周期的M-CDGT變換核中的復(fù)指數(shù)改換為離散Hartley變換(DHT)中的變換核cas實(shí)函數(shù),從而提出了周期(有限長序列)的M-RDGT;利用雙正交分析法研究了M-RDGT與其逆變換(即多窗實(shí)值離散Gabor展開)中的窗函數(shù)雙正交關(guān)系式,并證明此關(guān)系式等同于M-RDGT的完備性條件；研究了利用窗函數(shù)雙正交關(guān)系式求解窗函數(shù)的快速算法；研究了基于DHT的M-RDGT及其逆變換的快速算法；由于M-RDGT系數(shù)與M-CDGT系數(shù)之間關(guān)系就如同離散Fourier變換(DFT)系數(shù)與DHT系數(shù)之間關(guān)系(非常簡單的代數(shù)關(guān)系),因此M-RDGT快速算法也提供了一種快速有效地計(jì)算M-CDGT方法。提出了超長(或無限長)序列M-RDGT及其快速算法。在有限長系列(周期)的M-RDGT中,窗函數(shù)長度與待分析序列長度必須相同,對于超長序列的M-RDGT計(jì)算,無疑將大幅增加求解窗函數(shù)所需的計(jì)算量及存儲空間,有時甚至導(dǎo)致求解數(shù)值不穩(wěn)定。為了使得窗函數(shù)長度不隨待分析序列長度變化,即用長度較短的窗函數(shù)分析超長甚至無限長待分析序列,本文在有限長序列M-RDGT基礎(chǔ)上研究了超長序列M-RDGT及其快速算法,推導(dǎo)了超長序列M-RDGT在滿足完備性條件下新的窗函數(shù)雙正交關(guān)系式。提出了多抽樣率快速并行實(shí)現(xiàn)多窗實(shí)值離散Gabor變換方法。借助于多抽樣率數(shù)字濾波器組的分析與綜合基本原理和多窗離散Gabor展開與變換中分析(求變換系數(shù))與綜合(展開即信號重建)原理的相似性,設(shè)計(jì)了一種并行多抽樣率分析與綜合卷積器組來實(shí)現(xiàn)多窗實(shí)值離散Gabor展開與變換。所設(shè)計(jì)的分析和綜合卷積組中的每一并行通道具有一致的結(jié)構(gòu)并能夠利用快速DHT算法減小計(jì)算量。每一并行通道計(jì)算復(fù)雜性只決定于輸入離散信號的長度及Gabor頻率抽樣點(diǎn)數(shù),不會隨M-RDGT過抽樣率及窗數(shù)增加而增大,因此,每一并行通道的計(jì)算復(fù)雜性非常小。最后,本文對包含沖激函數(shù)的正弦函數(shù)序列、指數(shù)衰減正弦類瞬變序列以及Apnea-ECG數(shù)據(jù)庫中的心電(ECG)序列進(jìn)行了M-RDGT和時頻譜計(jì)算實(shí)驗(yàn),實(shí)驗(yàn)結(jié)果表明所提出的M-RDGT提供了一種快速有效的方法分析和展示包含有多個或時變頻率分量信號的動態(tài)時頻內(nèi)容。
[Abstract]:Gabor transform is a time-frequency analysis method is one of the important and widely used in the detection of non-stationary signal analysis and processing. However, due to the Heisenberg uncertainty principle constraints, the traditional single window Gabor transform time-frequency locality (or time-frequency resolution) is very limited. The width of window and window function comprehensive analysis is fixed, obtained by the single window Gabor transform spectrum time resolution and frequency resolution is fixed, and at the same time can not be all good, the two is contradictory. The wide window will produce high frequency resolution but low time resolution spectrum; on the other hand, the use of narrow window will have a high time resolution but when the spectrum of low frequency resolution. So the scholars in the theoretical framework is proposed based on the multi window complex valued discrete Gabor transform (M-CDGT), when the spectrum was obtained through the combination of multi window case, can effectively improve the time-frequency representation of the fine The degree of M-CDGT. But the research is still not perfect and thorough, in order to expand the research of multi window discrete Gabor transform, this paper studied the biorthogonal multi window real valued discrete Gabor transform (M-RDGT) method. The main research contents and innovations are as follows: the finite length sequence of M-RDGT and its fast algorithm. The cycle of the M-CDGT kernel the complex index change for the discrete Hartley transform (DHT) transform kernel CAS real function, thus put forward the cycle (finite length sequence) of M-RDGT; on M-RDGT and inverse analysis method using two orthogonal (i.e. multi window real valued discrete Gabor expansion) biorthogonal window function in the relationship, and to prove the completeness this relation is equivalent to M-RDGT; was studied by using the fast algorithm of Bi orthogonal relation window function solving window function; Study on fast algorithm of M-RDGT and the DHT inverse transformation based on M-RDGT and M-C; the coefficient The relationship between DGT coefficients as discrete Fourier transform (DFT) the relationship between the coefficient and DHT coefficient (algebraic relations very simple), so M-RDGT algorithm also provides a fast and effective method of calculating M-CDGT. Put forward long (or infinite) sequence of M-RDGT and the fast algorithm. In the limited series (long period) in M-RDGT, the window function and the length of stay of the length of the sequence must be the same, for long sequences of the M-RDGT calculation, will undoubtedly increase the amount of computation and storage space required for window function, sometimes even lead to numerical instability. In order to make the window length with sequence length changes to be analyzed, using window function analysis of long short length or even infinite long sequence analysis, this study of the long M-RDGT sequence and its fast algorithm in a finite length sequence of M-RDGT based on the deduced sequence of M-RDGT in length to meet the completeness Bi orthogonal relation window function under the new conditions. Proposed multirate fast parallel implementation of multi window real valued discrete Gabor transform method. With the analysis of analysis of multirate digital filter banks and the basic principle of comprehensive and multi window discrete Gabor expansion and transform (for transform coefficients) and synthesis (namely signal reconstruction) the similarity principle, a parallel multi rate sampling analysis and comprehensive convolver group to achieve multi window real valued discrete Gabor expansion and transform design. Each parallel channel analysis and comprehensive design of convolution group has the same structure and can use the fast DHT algorithm to reduce the computation of each parallel channel is calculated. The complexity depends only on the input length and Gabor frequency sampling points of discrete signals, not with the M-RDGT sampling rate and window number increases, therefore, the computational complexity of each parallel channel is very small. Finally, the The sequence contains sine impulse function, exponential sinusoidal transient sequences in the Apnea-ECG database and ECG (ECG) sequence of M-RDGT and spectrum calculation experiment, experimental results show that the proposed M-RDGT provides a fast and effective method of analysis and display with the dynamic variable frequency component signal multiple or when the frequency content.

【學(xué)位授予單位】：安徽大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：TN911.6
，

本文編號：1703591